Unifying Direct and Indirect Learning for Safe Control of Linear Systems

This paper develops learning-enabled safe controllers for linear systems subject to system uncertainties and bounded disturbances. Given the disturbance zonotope, the databased closed-loop dynamics (CLDs) are first characterized using a matrix zonotope (MZ), and refined through several steps to yield a constrained matrix zonotope (CMZ). This refinement is achieved by introducing conformal equality constraints that eliminate incompatible disturbance realizations. More precisely, prior knowledge and observed data are used separately to construct CMZ representations of disturbance sequences that conform to both data and prior knowledge, and are intersected by the initial MZ of the disturbance sequence, producing a refined CMZ. This approach reduces conservatism. To further reduce the conservativeness, we unify open-loop learning with closed-loop learning by presenting a novel set-membership identification method that models open-loop dynamics as a CMZ. The prior knowledge serves as an initial feasible open-loop model set (FOLMS) of this CMZ, which is refined into a posterior set whenever new informative online data becomes available. This posterior FOLMS then adaptively replaces the prior knowledge set employed in the disturbance elimination of the closed-loop learning process. The resulting refined parameterized set of CLD is subsequently leveraged to directly and adaptively learn a controller that robustly enforces safety. Toward this goal, we formulate a linear programming problem that guarantees λcontractiveness of a polyhedral safe set. A simulation example is provided to validate the effectiveness of the proposed approach and support the theoretical results.